Angular Velocity Calculator
Angular Velocity ω (rad/s)6.2832
Rotational Speed (rpm)60.00
Linear Velocity v (m/s)3.1416
The Angular Velocity Calculator measures how fast something rotates, expressed in radians per second. Work from an angular displacement over a time interval (ω = θ/t) or convert a rotational speed in revolutions per minute (ω = 2π·rpm/60). Supply a radius and the calculator also returns the linear, or tangential, speed v = ω·r of a point on the rotating body. Results are shown in rad/s, rpm, and m/s together.
Formula
ω = θ / t; ω = 2π · rpm / 60; v = ω · r
- ω
- Angular velocity (radians per second)
- θ
- Angular displacement (radians)
- t
- Elapsed time (seconds)
- rpm
- Rotational speed (revolutions per minute)
- r
- Radius from the axis to the point (metres)
How it works
- Choose a mode: enter an angular displacement in radians divided by a time in seconds, or enter a speed in RPM.
- Optionally enter a radius so the calculator can convert rotation into the linear speed of a point at that distance from the axis.
- The tool computes ω in rad/s, the equivalent rpm, and the tangential speed v = ω·r in metres per second.
Worked example
A wheel spinning at 60 rpm with a 0.5 m radius.
- ω = 2π × 60 / 60 = 2π = 6.2832 rad/s.
- v = ω × r = 6.2832 × 0.5.
- v = 3.1416 m/s.
Angular velocity ≈ 6.2832 rad/s, linear speed ≈ 3.1416 m/s.
Frequently asked questions
- How do I convert RPM to radians per second?
- Multiply the RPM value by 2π and divide by 60, because one revolution is 2π radians and one minute is 60 seconds. So 60 rpm equals 2π ≈ 6.283 rad/s.
- What is the difference between angular and linear velocity?
- Angular velocity (ω) measures rotation rate in rad/s and is the same for every point on a rigid body. Linear velocity (v = ω·r) is the straight-line speed of a specific point and grows with its distance r from the axis.
- Why use radians instead of degrees?
- Radians make the link between rotation and arc length direct: arc = r·θ only holds when θ is in radians. That is why v = ω·r works cleanly with ω in rad/s rather than degrees per second.
- Can angular velocity be negative?
- Yes. A negative value simply indicates rotation in the opposite direction (for example clockwise versus counter-clockwise). The magnitude still represents how fast the object turns.